Estimation of unknown states for an electric water heater with thermal stratification and use of same in demand response and condition-based maintenance

ABSTRACT

A water heater that includes a cylindrical storage tank, at least one heating element, and at least one temperature sensor is modeled using a one-dimensional model that includes: a vertical stack of disks representing the water volume in the cylindrical storage tank, and a stack of annular segments surrounding the vertical stack of disks. The stack of annular segments represents the cylindrical wall of the cylindrical storage tank. The one-dimensional model may be used by a condition-based maintenance system comprising an electronic data processing device configured to detect a failure mode present in the water heater based on an output of the water heating model component. Some illustrative failure modes include insulation disturbance, heating element failure, excessive sediment buildup, or a drip tube rupture.

This application claims the benefit of U.S. Provisional Application No.62/095,593 filed Dec. 22, 2014 and titled “ESTIMATION OF UNKNOWN STATESFOR AN ELECTRIC WATER HEATER WITH THERMAL STRATIFICATION AND USE OF SAMEIN DEMAND RESPONSE AND CONDITION-BASED MAINTENANCE”. U.S. ProvisionalApplication No. 62/095,593 filed Dec. 22, 2014 is incorporated herein byreference in its entirety.

BACKGROUND

The following relates to the water heater arts, water heater controlarts, water heater maintenance arts, and related arts.

Water heaters are ubiquitous appliances in residential and commercialsettings, used to provide hot water for washing, cleaning, laundryprocessing, industrial processes, and so forth. A typical electric waterheater includes a water storage tank with one or more heating elements,typically at upper and lower positions. Cold water enters near thebottom of the water storage tank via a cold water feed pipe, and isheated by the heating elements. Heated water loses density, causing itto tend to rise upward, and this flow pattern is reinforced by entry ofcold water near the bottom of the tank and extraction of hot water fromthe top of the tank. A gas water heater operates similarly, with theresistive electrical heating elements being replaced by a gas burnerusually located near the bottom of the water storage tank. In eithercase, temperature control is typically achieved by a simplethermostat-based controller that applies heat when the water temperaturein the storage tank falls below a deadband minimum and turns off theheater (gas or electric) when the water temperature rises above adeadband maximum. Within the deadband the heater setting remainsunchanged, producing a temperature cycling within the deadband (possiblywith some overshoot and/or undershoot) about a temperature set pointlocated at about the middle of the deadband. This type of controladvantageously leverages thermal hysteresis to reduce the on/off cyclingof the heating element. Water temperature is usually set by adjustingthe set point, with the deadband limits defined relative to the setpoint (e.g., ±2° C. above/below the thermostat set point).

Recognizing that water heaters in a building, city, or region representa large distributed thermal energy storage reservoir, there has beeninterest in leveraging aggregations of water heaters as energy storagedevices to provide demand response, in which the electrical load of theelectric grid is matched with electrical generation. (By comparison,conventionally the power generation is adjusted to match load, forexample by bringing ancillary power generators online/offline as neededto match load). By way of illustration, to perform load shedding thewater heater operation can be curtailed during peak energy usageperiods, with hot water continuing (for a time) to be available from thehot water tank. As another illustration, in frequency control the loadis adjusted at a higher frequency, typically on the order of seconds, inaccord with an Automatic Generation Control (AGC) signal to maintain thegrid frequency.

To perform demand response, especially at higher frequencies such asthose required for AGC-based frequency control, the water heaterstypically must be controlled remotely, for example by retrofitting thewater heater with a remotely operable load controller (or, in the caseof a new water heater, including such a load controller as an originalmanufacturer component). Also, the demand response must be balancedagainst the traditional function of water heaters: to provide hot water(which limits the time that the water heater can be kept off), as wellas safety considerations such as not overloading the electricalcircuits, or generating water that is scalding hot (which limits thetime the water heater can be kept heating). To balance theseconsiderations, it is useful to provide feedback to the aggregatecontroller, such as the water temperature in the storage tank,instantaneous water heater power consumption, or so forth.

BRIEF SUMMARY

In some illustrative embodiments disclosed as illustrative examplesherein, a water heater control system is disclosed for controlling awater heater that includes a vertically oriented cylindrical waterstorage tank having a cylindrical wall, at least one heating elementarranged to heat water in the water storage tank, and at least onetemperature sensor arranged to measure water temperature in the waterstorage tank. The electric water heater control system comprises: a loadcontroller comprising an electronic data processing device configured tooperate the water heater including operating the at least one heatingelement based on temperature readings provided by the at least onetemperature sensor to control water temperature of water in the waterstorage tank; and a water heater modeling component comprising anelectronic data processing device configured to model the water heaterusing a one-dimensional model that includes: a vertical stack of disksrepresenting the water volume in the cylindrical water storage tank, anda stack of annular segments surrounding the vertical stack of diskswherein the stack of annular segments represents the cylindrical wall ofthe cylindrical water storage tank. In some embodiments theone-dimensional model comprises coupled differential equationsincluding: (1) for each disk of the vertical stack of disks representingthe water volume in the cylindrical water storage tank, a differentialequation expressing the time derivative of the temperature of the diskestimated by the one-dimensional model; and (2) for each annular segmentof the stack of annular segments representing the cylindrical wall ofthe cylindrical water storage tank, a differential equation expressingthe time derivative of the temperature of the annular segment estimatedby the one-dimensional model.

In some illustrative embodiments disclosed as illustrative examplesherein, a system includes a water heater, a load controller, anaggregation demand response dispatch engine, and a condition-basedmaintenance system. The water heater includes a water storage tank, atleast one heating element arranged to heat water in the water storagetank, and at least one temperature sensor arranged to measure watertemperature in the water storage tank. The load controller comprises anelectronic data processing device configured to operate the water heaterincluding operating the at least one heating element based ontemperature readings provided by the at least one temperature sensor tocontrol water temperature of water in the water storage tank. Theaggregation demand response dispatch engine comprises an electronic dataprocessing device configured to send demand response commands to loadcontrollers of an aggregation of loads including the load controllerconfigured to operate the water heater. The load controller is furtherconfigured to operate the water heater in accord with demand responsecommands received from the aggregation demand response dispatch engine.A condition-based maintenance system comprises an electronic dataprocessing device configured to detect a failure mode present in thewater heater based on information including the temperature readingsprovided by the at least one temperature sensor and power input to thewater heater.

In some illustrative embodiments disclosed as illustrative examplesherein, a system comprises: an electrical load; a load controllercomprising an electronic data processing device configured to operatethe electrical load; an aggregation demand response dispatch enginecomprising an electronic data processing device configured to senddemand response commands to a loads aggregation including sending demandresponse commands to the load controller configured to operate theelectrical load, wherein the load controller is further configured tooperate the electrical load in accord with demand response commandsreceived from the aggregation demand response dispatch engine; and acondition-based maintenance system comprising an electronic dataprocessing device configured to detect a failure mode present in theelectrical load based on information also input to the load controller.In some embodiments, the condition-based maintenance system isconfigured to detect a failure mode present in the electrical load basedon information also input to the load controller including (i) readingsof at least one temperature sensor that measures temperature of a fluidvolume whose temperature is controlled by the load controller operatingthe electrical load and (ii) electrical power input to the electricalload. In some embodiments the electrical load is one of an electricwater heater and a heating, ventilation, and air conditioning (HVAC)system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 diagrammatically shows a demand response system employing anaggregation of electrical loads comprising illustrative water heaters.

FIG. 2 diagrammatically shows a side partial sectional view of one ofthe water heaters of the system of FIG. 1.

FIG. 3 diagrammatically shows a one-dimensional model of the waterheater of FIG. 2 including a vertical stack of disks representing thewater volume in the cylindrical water storage tank, and a stack ofannular segments surrounding the vertical stack of disks. The stack ofannular segments represents the cylindrical wall of the cylindricalwater storage tank.

FIG. 4 diagrammatically shows a process flow performed by acondition-based maintenance (CBM) component of the demand responsesystem of FIG. 1.

DETAILED DESCRIPTION

Disclosed herein are improved approaches for modeling a water heater foruse in advanced operations such as demand response and condition-basedmaintenance. These approaches recognize that monitoring water heateroperation by tracking temperature reading of the thermostat may notprovide sufficient information for some advanced operations such asdemand response. In order for electric water heaters to be effectivelyused as an energy storage device for demand response, the energy storedin the water heater should be tracked as a function of time. Theinternal temperature of water heater tank, typically as measured by thethermostat or an ancillary thermocouple or other temperature measuringdevice, is typically taken as the energy storage state metric in orderto infer available energy. This metric assumes that the water in thetank is well-mixed, so that the water temperature is uniform throughoutthe tank. However, it is recognized herein that vertically orientedcylindrical water storage tanks, of the type commonly used inresidential and commercial settings, may exhibit strong temperaturestratification along the vertical direction, with the temperature nearthe bottom of the tank usually being lower than the temperature near thetop of the tank. This vertical stratification is caused by theconfiguration of cold water feed and hot water draw lines: the coldwater is supplied near the bottom of the tank, while the the hot wateris drawn from the top of the tank. Vertical temperature stratificationalso results from differences in density between the cold water and holdwater—heating the water causes it to become less dense, leading thehotter water to tend to migrate upward through the tank. Verticaltemperature stratification can also result from the use of discreteheating elements that do not impart heat into the tank volume uniformly.

The vertical temperature stratification impacts the dynamic energybehavior of the water heater. To model the temperature in the watertank, a complex three-dimensional (3D) model may be employed thatincorporates computational fluid dynamics (CFD) methods to solve forthermal and mass transport dynamics. These methods are highlycomputationally expensive, and may be impractical for implementation inrelatively simple electronics such as those desired to be incorporatedinto a water heater controller. Another difficulty is that such 3Dmodeling typically requires substantial information to be input to themodel, which may be unavailable. Temperatures within the water storagetank are typically unavailable except at one or a two discrete points(e.g. one temperature reading per thermostat, with two thermostatsprovided in some conventional electric water heater designs). Similarly,water flow rates may be unavailable.

Attempts have been made to simplify the modeling by employing aone-dimensional water heater model. See Fanney et al., “The ThermalPerformance of Residential Electric Water Heaters Subjected to VariousOff-Peak Schedules”, Journal of Solar Energy Engineering, vol. 118 pp.73-80 (1996); Vettros et al., “Load Frequency Control by Aggregations ofThermally Stratified Electric Water Heaters”, Innovated Smart GridTechnologies (ISGT Europe), 2012 3^(rd) IEEE PES ISGT (IEEE 2012).However, these models do not account for three-dimensional effects, suchas thermal losses at the shell of the tank 40.

Disclosed herein are one-dimensional (1D) lumped parameter models thatoperate on only two tank wall measurements with a known heating input toapproximate the vertical temperature stratification of the water heater.The disclosed 1D models are suitably used to estimate internal watertemperatures of the tank, as well as the (unmeasured) water draw flowrate and temperature. The 1D models disclosed herein leverage thesubstantial symmetry about the vertical axis of the cylindrical waterstorage tank of a typical water heater by modeling the cylindrical watertank as a stack of disks (sub-cylinders) along the vertical axis of thecylindrical water storage tank. Additionally, the disclosed 1D waterheater models provide effective modeling of thermal behavior at theboundary of the tank 40 by including annular segments representing thetank wall, while still retaining the advantageous 1D formalism. Thisallows the use of wall temperature measurements to infer the internalunknown stratified dynamics.

The 1D model operating on limited inputs from the conventional waterheater thermostat(s) provides more accurate information about the energystored in the water heater tank at any given time, thus providing moreeffective demand response while ensuring the primary water heaterfunction of providing hot water is also safely and effectivelyperformed. The information provided by such a model can be leveraged toprovide more accurate information about the operational status of keycomponents of the water heater, thus providing information suitably usedto perform condition-based maintenance of the water heater.

With reference to FIG. 1, a demand response system is disclosed, whichleverages one or more (illustrative four) electric water heaters 20 of afacility 22 (residence, business, or so forth) to provide demandresponse in support of an electric grid managing entity 24. Because eachwater heater 20 individually provides a small amount of energy storageas compared with the electric grid, the water heaters 20 (possibly alongwith water heaters of other facilities, and/or along with otherelectrical devices capable of storing energy such as air conditions) areoperated as an aggregation by a demand response dispatch engine 26 inorder to coordinate operations of the water heaters (and optionallyother loads of the aggregation) to provide demand response services.More particularly, the dispatch engine 26 is suitably embodied by acomputer or other electronic data processing device communicating withload controllers 30 of the water heaters 20 or other electrical loads ofthe aggregation via a communication network 32 and programmed to receivedemand response instructions from the grid managing entity 24 (forexample, curtailment instruction in the case of a load sheddingoperation, or an AGC signal in the case of frequency regulation) and tosend control commands to the load controllers 30 to cause the loadcontrollers 30 to control their respective water heaters to implementthe demand response instruction. The communication network 32 may, byway of illustrative example, include wired or wireless Internet links,wired or wireless local area network (LAN) links, Bluetooth links,various combinations thereof, or so forth. As a more particularillustrative example, the demand response dispatch engine 26 may beconnected with the Internet and send control commands via the Internetto an IP address associated with a facility controller (not shown) ofthe facility 22 which retransmits the control commands to network accesspoints distributed through the facility 22 (possibly with sometranslation or other processing of the control commands at the facilitycontroller) via a wired or wireless local area network, and the accesspoints then transmit the control commands to the load controllers 30 viaBluetooth, wired connections, or another short-range communication link.These are merely illustrative examples. The load controllers 30 compriseelectronic data processing devices and may be variously embodied, forexample as microcontroller- or microprocessor-based controllers thatreplace the original manufacturer thermostat, or that augment operationof the original thermostat by operating a power relay connected with thepower feed to the water heater, or intercepting and modifying thetemperature input to the thermostat, or so forth. In some embodiments,the load controller 30 may be an original manufacturer component, e.g. athermostat originally designed to accept remote control commands.Communication between the grid managing entity 24 and aggregationdispatch controller 26 may be via similar pathways (Internet, LAN, etcetera). Additionally or alternatively, demand response instructions maybe conveyed from the grid managing entity 24 to the aggregation dispatchcontroller 26 manually, for example via telephone to a human data entryoperator of a computer embodying the dispatch controller 26.

Besides providing a large aggregate energy storage capacity for demandresponse operations, another advantage of operating the water heaters 20in an aggregation context is that this assists in ensuring thatindividual water heaters can deviate from the desired aggregate responsein order to fulfill other, possibly contradictory, demands on the waterheater, such as providing hot water or not overheating the water in thestorage tank. Such deviations may be uncoordinated and/or coordinated.As an illustrative example of an uncoordinated deviation, a water heatermay be commanded by the dispatch engine 26 to curtail energyusage—however, if the water in the water heater's storage tank fallsbelow a deadband minimum of the thermostat, the load controller 30controlling the water heater (or the original thermostat, depending onthe precise control configuration) may activate the heating elements ofthe water heater in deviation from the energy usage curtailment command.If the deviant water heater is only one member of a relatively largeaggregation of water heaters, then this uncoordinated deviationnonetheless will not significantly affect the demand responsecurtailment provided by the aggregation, and it allows the water heaterto perform its primary duty of providing hot water. As an example of acoordinated deviation, during a load shedding operation the dispatchengine 26 may receive state information regarding the states of thewater heaters 20 from their respective controllers 30, and based on thisinformation the dispatch engine 26 may instruct a water heater with alow amount of thermal energy stored in its tank (i.e. the water isrelatively cold) to operate in deviance from the curtailment command.This latter, coordinated approach requires more algorithmic complexityat the dispatch engine 26 in order to prioritize the curtailment amongstthe water heaters of the aggregation, but has the advantage that theprioritization can better balance the demand response (e.g. curtailment)against individual load needs.

Each load controller 30 operates in part based on information generatedby maintaining a dynamic one-dimensional (1D) model 33 of the waterheater 20 under control of the load controller 30. In FIG. 1, the 1Dwater heater model 33 is diagrammatically shown in the upper right insetof FIG. 1—however, it is to be understood that a separate instance ofthe model is executed at each load controller 30 to model the waterheater under control of that controller 30. (In an alternativeembodiment, it is contemplated to execute the 1D model 33 for each waterheater at the aggregation dispatch engine 26, based on inputs receivedfrom the load controller). The 1D water heater model 33 receives alimited number of inputs, and estimates the vertical temperaturedistribution through the volume of the water tank. For example, in oneillustrative embodiment, the 1D model 33 receives as input the powerreading for each heating element (of which there may be one or moreheating elements for a given water heater), the temperature reading forthe thermostat controlling each heating element, and an ambienttemperature reading (or assumed room temperature value if the roomcontaining the water heater has a well-controlled temperature). Theseinputs are readily obtained by interfacing with the thermostat(s) of thewater heater 20, by adding thermocouples or other temperaturemeasurement devices to the wall or skin of the water heater holdingtank, by employing a clamp-on ammeter or the like to measure electricalpower input to the heating elements, by adding a room thermometer (orthermocouple, et cetera), or so forth.

The output of the 1D model 33 for each controlled water heater 20 may beused in demand response applications to provide more accurate estimationof the thermal energy stored in the water tank as compared with a watertemperature reading provided by the original thermostat. However, thisis merely one possible application and others are contemplated. By wayof illustrative example, another suitable application of the 1D model 33is for estimating the operational condition of the modeled water heater20. In this condition-based maintenance application, the more detailedknowledge regarding the operational state of the water heater 20provided by the model 33 enables automated diagnosis of certain commonfailure modes such as partial insulation failure, heating elementfailure, excessive tank sediment build-up, or a rupture in the coldwater feed drip tube that directs cold water to the bottom of theholding tank. Such a failure diagnosis may be communicated to the demandresponse dispatch engine 26 so that the dispatch engine can take thediagnosed failure into account in providing demand response services tothe grid operator 24 (for example, by not using the apparently failedwater heater to provide demand response). Additionally or alternatively,the failure diagnosis may be communicated to a facility maintenanceentity 34 having responsibility for maintaining the water heaters 20 ofthe facility 22. The facility maintenance entity 34 may, for example,comprise a front-desk computer staffed by a building maintenancedepartment and having a notifications component via which buildingmaintenance personnel may be notified of the diagnosed failure, and/orthe facility maintenance entity 34 may comprise a cellular telephone ornetworked tablet device carried by the building maintenance person andconfigured to push notifications to the user.

With reference to FIGS. 2 and 3 an illustrative example of an electricwater heater 20 is described (FIG. 2), along with a suitable embodimentof a 1D water heater model 33 (FIG. 3) suitably modeling the waterheater of FIG. 2. The illustrative electric water heater 20 includes awater storage (or holding) tank 40 containing hot water (or water to beheated), an upper resistive heating element 42 and a lower resistiveheating element 44 that can be electrically energized to heat water inthe tank 40, a cold water inlet 46 including a drip tube 48 that extendsdownward to deliver cold water near the bottom of the tank 40, and a hotwater draw line 50 positioned to draw hot water from at or near the topof the tank 40. The upper heating element 42 is controlled by an upperthermostat 52, and the lower heating element 44 is controlled by a lowerthermostat 54. In a typical configuration, the upper thermostat 52includes a set point adjustment knob (not shown) via which a user mayadjust the temperature set point of the water heater 20, and the lowerthermostat 54 is operatively linked with the upper thermostat 52.

The illustrative water heater 20 is of the vertically orientedcylindrical design, in which the storage tank 40 has the general shapeof a cylinder defining a vertical tank (cylinder) axis 56. Although thetank 40 has the general shape of a vertically oriented cylinder, variousdeviations from the cylindrical shape may be present, such asillustrative rounded top and bottom portions, features such as theresistive heating elements 42, 44 that may break perfect cylindricalsymmetry, or so forth. A drain valve 58 is also provided to enable waterin the tank 40 to be drained out for maintenance, transport ordecommissioning of the water heater 20, or for other purposes.

Also diagrammatically depicted in FIG. 2 are some failure modes that canbe detected based on parameters estimated using the model 33. Anillustrative region of corrosion 60 on the skin of the water heater tank40, for example due to repeated exposure to water from the drain valve52 or another source, may produce an insulation disturbance thatincreases thermal losses from the tank 40 and reduces water heaterefficiency. The illustrative lower heating element 44 includes somefouling or build-up 62 that can reduce its ability to transfer heat intothe water, thus lowering the heating efficiency. The bottom of the tank40 also has an excessive sediment build-up 64 that reduces the actualwater capacity of the tank 40 and may be indicative of a more seriousproblem such as corrosion of the inner lining of the tank 40. Stillfurther, the drip tube 48 has developed a rupture 66 generating a flowpath 68 via which cold water can pass directly to the hot water drawline 50 without being significantly heated by the heating elements 42,44.

With continuing reference to FIG. 2, the illustrative load controller 30that controls the illustrative water heater 20 is mounted on the skin oroutside wall of the water storage tank 40. This is merely anillustrative example, and the load controller may in general beinstalled at various locations, for example wall-mounted and connectedwith the water heater by suitable cabling or so forth. The illustrativeload controller 30 includes a microprocessor or microcontroller andassociated components (e.g. electronic memory, analog-to-digitalconverters for reading thermocouples or other analog inputs, a USB portor other digital interface for receiving digital inputs or outputtingdigital outputs, a digital-to-analog converter for outputting an analogcontrol signal, a WiFi interface, Ethernet interface, or othernetworking interface, various combinations, sub-sets, or so forth of theforegoing, et cetera) programmed or configured to execute a water heatercontrol algorithm based on received remote inputs such as demandresponse control commands from the dispatch engine 26 (see FIG. 1). Inperforming such control operations, the load controller 30 executes the1D water heater model 33 to model the current operational state based oninputs including: the power delivered to (or consumed by) the heatingelements 42, 44; the temperature readings of the thermostats 52, 54; andan assumed or measured room (ambient) temperature. The water heaterstate information provided by the executing 1D water heater model 33 mayalso be used by a condition based maintenance algorithm executed by theload controller 30 to detect various failure modes, such as illustrativefailure modes 60, 62, 64, 66, 68. In alternative embodiments, the 1Dmodeling algorithm and/or the failure mode detection algorithm mayexecute at the computer embodying the dispatch engine 26 or elsewhere.

With reference to FIG. 3, an illustrative embodiment of the 1D waterheater model 33 is described, which is suitable for modeling theillustrative water heater 20 shown in FIG. 2. The 1D water heater model33 utilizes two external temperature measurements from the thermostats52, 54 (or, alternatively, from similarly located thermostats mounted tothe wall of the tank 40), the heating input to the heating elements 42,44 (assumed to correspond to the electrical power input to the heatingelements 42, 44), and an ambient temperature measurement (or assumedambient temperature value). Estimates all the other internal andexternal temperatures are generated by the 1D model 33, along withestimates of the water draw mass flow rate and inlet water temperaturewith disturbance estimators. The 1D model 33 is a one-dimensional lumpedmodel structure of the internal and external temperature dynamics of thewater heater 20, in which the water tank 40 is divided into N finitelumps or disks. In the illustrative model 33 of FIG. 3, N=9 includingdisks representing the top and bottom of the tank.

More particularly, the illustrative 1D water heater model 33 of FIG. 3includes a bottom disk 1 representing the bottom of the tank 40 and,running successively upward along the vertical cylinder axis 56, sevensuccessive disks 2, 3, 4, 5, 6, 7, 8 each representing a successivelyhigher disk of water in the stack of disks representing the volume ofthe water holding tank 40, topped at the highest elevation along theaxis 56 by a top disk 9 representing the top of the tank 40. The wallsof the cylindrical holding tank 40 between the top disk 1 and bottomdisk 9 are also modeled, being represented by seven annular elements 10,11, 12, 13, 14, 15, 16 at successively higher elevation and surroundingcorresponding disks 2, 3, 4, 5, 6, 7, 8. Said another way, the lumpedwall segment 10 is an annular segment surrounding the disk 2; the lumpedwall segment 11 is an annular segment surrounding the disk 3; the lumpedwall segment 12 is an annular segment surrounding the disk 4; the lumpedwall segment 13 is an annular segment surrounding the disk 5; the lumpedwall segment 14 is an annular segment surrounding the disk 6; the lumpedwall segment 15 is an annular segment surrounding the disk 7; and thelumped wall segment 16 is an annular segment surrounding the disk 8.These annular segments enable accurate modeling of the impact of thermallosses at the walls of the water storage tank 40, while retaining theone-dimensional formalism so as to substantially reduce computationalcomplexity as compared with three-dimensional modeling approaches.

Heat injected by the upper heating element 42 is represented by a heatinput H1 into disk 7, and heat injected by the lower heating element 44is represented by a heat input H2 into disk 3. The temperature measuredby the upper temperature sensor (e.g. thermostat 52) is indicated as atemperature reading M16 of the annular wall element 16, while thetemperature measured by the lower temperature sensor (e.g. thermostat54) is indicated as a temperature reading M13 of the annular wallelement 13. The cold water is assumed to be input (via the drip tube 48)into the lowest disk 2 and to have a temperature denoted as T_(in). Thehot water drawn is assumed to have the temperature of the uppermost disk8. It will be appreciated that the number of disk/annulus divisions, andthe locations of the heat inputs and temperature readings respective tothose divisions, is suitably chosen to provide a desired spatialresolution in the vertical direction and to accurately model the actualpositions of the heating elements and temperature sensors in the waterheater being modeled.

With continuing reference to FIGS. 2 and 3, the dynamic behavior of thewater heater 20 as represented by the 1D model 33 of FIG. 3 can berepresented mathematically by a one-dimensional model including: (1) foreach disk of the vertical stack of disks representing the water volumein the cylindrical water storage tank, a differential equationexpressing the time derivative of the temperature of the disk estimatedby the one-dimensional model; and (2) for each annular segment of thestack of annular segments representing the cylindrical wall of thecylindrical water storage tank, a differential equation expressing thetime derivative of the temperature of the annular segment estimated bythe one-dimensional model. In an illustrative example, the 1D model 33of FIG. 3 is suitably represented by the following set of DifferentialEquations (1)-(20):

$\begin{matrix}{\frac{d{\hat{T}}_{1}}{d\; t} = {\frac{\left( {{\hat{T}}_{2} - {\hat{T}}_{1}} \right)A}{m_{1}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{10} - {\hat{T}}_{1}} \right)A}{m_{1}{c_{wall}\left( {\frac{\Delta\; y_{w}}{2k_{w}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{\left( {A_{{wall},1} + A_{b}} \right)U}{m_{1}c_{wall}}\left( {{\hat{T}}_{1} - T_{amb}} \right)} + {L_{1}{\overset{\_}{T}}_{10}} + {L_{2}{\overset{\_}{T}}_{13}}}} & (1) \\{\frac{d\;{\hat{T}}_{2}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{2}c_{w}}\left( {{\hat{T}}_{3} - {\hat{T}}_{2}} \right)} + \frac{\left( {{\hat{T}}_{1} - {\hat{T}}_{2}} \right)A}{m_{2}{c_{w}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + {\frac{\hat{\overset{.}{m}}}{m_{2}}\left( {{\hat{T}}_{in} - {\hat{T}}_{2}} \right)} - \frac{\left( {{\hat{T}}_{2} - {\hat{T}}_{10}} \right)A}{m_{2}{c_{w}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + {L_{3}{\overset{\_}{T}}_{10}} + {L_{4}{\overset{\_}{T}}_{13}}}} & (2) \\{\frac{d{\hat{T}}_{3}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{3}c_{w}}\left( {{\hat{T}}_{2} - {2{\hat{T}}_{3}} + {\hat{T}}_{4}} \right)} + {\frac{\hat{\overset{.}{m}}}{m}\left( {{\hat{T}}_{2} - {\hat{T}}_{3}} \right)} - \frac{\left( {{\hat{T}}_{3} - {\hat{T}}_{11}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{\eta\; P}{m_{4}c_{w}}u_{1}} + {L_{5}{\overset{\_}{T}}_{10}} + {L_{6}{\overset{\_}{T}}_{13}}}} & (3) \\{\frac{d{\hat{T}}_{4}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{4}c_{w}}\left( {{\hat{T}}_{3} - {2{\hat{T}}_{4}} + {\hat{T}}_{5}} \right)} + {\frac{\hat{\overset{.}{m}}}{m_{4}}\left( {{\hat{T}}_{3} - {\hat{T}}_{4}} \right)} - \frac{\left( {{\hat{T}}_{4} - {\hat{T}}_{12}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{{\overset{.}{m}}_{cf}}{m_{4}}\left( {{\hat{T}}_{3} - {2{\hat{T}}_{4}} + {\hat{T}}_{5}} \right)} + {L_{7}{\overset{\_}{T}}_{10}} + {L_{8}{\overset{\_}{T}}_{13}}}} & (4) \\{\frac{d\; T_{5}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{5}c_{w}}\left( {{\hat{T}}_{4} - {2{\hat{T}}_{5}} + {\hat{T}}_{6}} \right)} + {\frac{\hat{\overset{.}{m}}}{m_{5}}\left( {{\hat{T}}_{4} - {\hat{T}}_{5}} \right)} - \frac{\left( {{\hat{T}}_{5} - {\hat{T}}_{13}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{{\overset{.}{m}}_{cf}}{m_{5}}\left( {{\hat{T}}_{4} - {2{\hat{T}}_{5}} + {\hat{T}}_{6}} \right)} + {L_{9}{\overset{\_}{T}}_{10}} + {L_{10}{\overset{\_}{T}}_{13}}}} & (5) \\{\frac{d{\hat{T}}_{6}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{6}c_{w}}\left( {T_{5} - {2T_{6}} + T_{7}} \right)} + {\frac{\hat{\overset{.}{m}}}{m_{6}}\left( {{\hat{T}}_{5} - {\hat{T}}_{6}} \right)} - \frac{\left( {{\hat{T}}_{6} - {\hat{T}}_{14}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{\eta\; P}{m_{6}c_{w}}u_{2}} + {\frac{{\overset{.}{m}}_{cf}}{m_{6}}\left( {{\hat{T}}_{5} - {2{\hat{T}}_{6}} + {\hat{T}}_{7}} \right)} + {L_{11}{\overset{\_}{T}}_{10}} + {L_{12}{\overset{\_}{T}}_{13}}}} & (6) \\{\frac{d{\hat{T}}_{7}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{7}c_{w}}\left( {{\hat{T}}_{6} - {2{\hat{T}}_{7}} + {\hat{T}}_{8}} \right)} + {\frac{\hat{\overset{.}{m}}}{m_{7}}\left( {{\hat{T}}_{6} - {\hat{T}}_{7}} \right)} - \frac{\left( {{\hat{T}}_{7} - {\hat{T}}_{15}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{{\overset{.}{m}}_{cf}}{m_{7}}\left( {{\hat{T}}_{6} - {2{\hat{T}}_{7}} + {\hat{T}}_{8}} \right)} + {L_{13}{\overset{\_}{T}}_{10}} + {L_{14}{\overset{\_}{T}}_{13}}}} & (7) \\{\frac{d{\hat{T}}_{8}}{d\; t} = {{\frac{k_{w}}{\Delta\; x_{w}m_{8}c_{w}}\left( {{\hat{T}}_{7} - {2{\hat{T}}_{8}} + {\hat{T}}_{9}} \right)} + {\frac{\hat{\overset{.}{m}}}{m_{8}}\left( {{\hat{T}}_{7} - {\hat{T}}_{8}} \right)} - \frac{\left( {{\hat{T}}_{8} - {\hat{T}}_{16}} \right)A}{\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} + {\frac{{\overset{.}{m}}_{cf}}{m_{8}}\left( {{\hat{T}}_{7} - {\hat{T}}_{8}} \right)} + {L_{15}{\overset{\_}{T}}_{10}} + {L_{16}{\overset{\_}{T}}_{13}}}} & (8) \\{\frac{d{\hat{T}}_{9}}{d\; t} = {\frac{\left( {{\hat{T}}_{9} - {\hat{T}}_{8}} \right)A}{m_{9}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{16} - {\hat{T}}_{9}} \right)A}{m_{9}{c_{wall}\left( {\frac{\Delta\; y_{w}}{2k_{w}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{\left( {A_{{wall},9} + A_{t}} \right)U}{m_{9}c_{wall}}\left( {{\hat{T}}_{9} - {\hat{T}}_{amb}} \right)} + {L_{17}{\overset{\_}{T}}_{10}} + {L_{18}{\overset{\_}{T}}_{13}}}} & (9) \\{\frac{d{\hat{T}}_{10}}{d\; t} = {\frac{\left( {{\hat{T}}_{2} - {\hat{T}}_{10}} \right)A}{m_{10}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{1} - {2{\hat{T}}_{10}} + T_{11}} \right)A}{m_{10}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{10}c_{wall}}\left( {{\hat{T}}_{10} - {\hat{T}}_{amb}} \right)} + {L_{19}{\overset{\_}{T}}_{10}} + {L_{20}{\overset{\_}{T}}_{13}}}} & (10) \\{\frac{d{\hat{T}}_{11}}{d\; t} = {\frac{\left( {{\hat{T}}_{3} - {\hat{T}}_{11}} \right)A}{m_{11}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{10} - {2{\hat{T}}_{11}} + T_{12}} \right)A}{m_{11}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{11}c_{wall}}\left( {{\hat{T}}_{11} - T_{amb}} \right)} + {L_{21}{\overset{\_}{T}}_{10}} + {L_{22}{\overset{\_}{T}}_{13}}}} & (11) \\{\frac{d{\hat{T}}_{12}}{d\; t} = {\frac{\left( {{\hat{T}}_{4} - {\hat{T}}_{12}} \right)A}{m_{12}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{11} - {2{\hat{T}}_{12}} + T_{13}} \right)A}{m_{12}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{12}c_{wall}}\left( {{\hat{T}}_{12} - T_{amb}} \right)} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (12) \\{\frac{d{\hat{T}}_{13}}{d\; t} = {\frac{\left( {{\hat{T}}_{5} - {\hat{T}}_{13}} \right)A}{m_{13}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{12} - {2{\hat{T}}_{13}} + T_{14}} \right)A}{m_{13}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{13}c_{wall}}\left( {{\hat{T}}_{13} - T_{amb}} \right)} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (13) \\{\frac{d{\hat{T}}_{14}}{d\; t} = {\frac{\left( {{\hat{T}}_{6} - {\hat{T}}_{14}} \right)A}{m_{14}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{13} - {2{\hat{T}}_{14}} + T_{15}} \right)A}{m_{14}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{14}c_{wall}}\left( {{\hat{T}}_{14} - T_{amb}} \right)} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (14) \\{\frac{d{\hat{T}}_{15}}{d\; t} = {\frac{\left( {{\hat{T}}_{7} - {\hat{T}}_{15}} \right)A}{m_{15}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{14} - {2{\hat{T}}_{15}} + T_{16}} \right)A}{m_{15}{c_{wall}\left( {\frac{\Delta\; y_{s}}{2k_{s}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{15}c_{wall}}\left( {{\hat{T}}_{15} - {\hat{T}}_{amb}} \right)} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (15) \\{\frac{d{\hat{T}}_{16}}{d\; t} = {\frac{\left( {{\hat{T}}_{8} - {\hat{T}}_{16}} \right)A}{m_{16}{c_{wall}\left( {\frac{\Delta\; x_{w}}{2k_{w}} + \frac{\Delta\; x_{s}}{2k_{s}}} \right)}} + \frac{\left( {{\hat{T}}_{9} - {\hat{T}}_{16}} \right)A}{m_{16}{c_{wall}\left( {\frac{\Delta\; y_{w}}{2k_{w}} + \frac{\Delta\; y_{s}}{2k_{s}}} \right)}} - {\frac{A_{{wall},1}U}{m_{16}c_{wall}}\left( {{\hat{T}}_{16} - T_{amb}} \right)} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (16) \\{\frac{d\hat{\overset{.}{m}}}{d\; t} = {{\hat{m}}_{2} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (17) \\{\frac{d{\hat{\overset{.}{m}}}_{2}}{d\; t} = {{{- {\hat{\alpha}}_{1}}{\hat{m}}_{1}} - {\alpha_{2}{\hat{m}}_{2}} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (18) \\{\frac{d{\hat{T}}_{in}}{d\; t} = {{\hat{T}}_{f} + {L_{23}{\overset{\_}{T}}_{10}} + {L_{24}{\overset{\_}{T}}_{13}}}} & (19) \\{\frac{d{\hat{T}}_{f}}{d\; t} = {{{- \alpha_{3}}{\hat{T}}_{in}} - {\alpha_{4}{\hat{T}}_{f}} + {L_{25}{\overset{\_}{T}}_{10}} + {L_{26}{\overset{\_}{T}}_{13}}}} & (20)\end{matrix}$In these Differential Equations, the following symbols are used:

A is the lateral area of each of the disks 2, 3, 4, 5, 6, 7, 8;

T_(amb) is the ambient temperature;

{circumflex over (T)}_(in) is the estimated temperature for cold waterdelivered into disk 2;

{circumflex over (T)}₁ is the estimated temperature for the bottom ofthe tank (disk 1);

{circumflex over (T)}₂ is the estimated temperature for water in thedisk 2;

{circumflex over (T)}₃ is the estimated temperature for water in thedisk 3;

{circumflex over (T)}₄ is the estimated temperature for water in thedisk 4;

{circumflex over (T)}₅ is the estimated temperature for water in thedisk 5;

{circumflex over (T)}₆ is the estimated temperature for water in thedisk 6;

{circumflex over (T)}₇ is the estimated temperature for water in thedisk 7;

{circumflex over (T)}₈ is the estimated temperature for water in thedisk 8;

{circumflex over (T)}₈ is also the estimated temperature for drawn hotwater;

{circumflex over (T)}₉ is the estimated temperature for the top of thetank (disk 9);

{circumflex over (T)}₁₀ is the estimated temperature for the annularwall segment 10;

{circumflex over (T)}₁₁ is the estimated temperature for the annularwall segment 11;

{circumflex over (T)}₁₂ is the estimated temperature for the annularwall segment 12;

{circumflex over (T)}₁₃ is the estimated temperature for the annularwall segment 13;

{circumflex over (T)}₁₄ is the estimated temperature for the annularwall segment 14;

{circumflex over (T)}₁₅ is the estimated temperature for the annularwall segment 15;

{circumflex over (T)}₁₆ is the estimated temperature for the annularwall segment 16;

{dot over ({circumflex over (m)})} is the estimated water draw flow rate(mass per unit time);

U is the heat transfer coefficient between the shell and theenvironment;

T is the error between measured and estimated temperatures;

L is the observer design gain, which is a design constant that multiplesthe error between the measurement and the estimation of the two shelltemperatures;

Δx is the thickness of each disk in the horizontal direction;

Δy is the thickness of each disk in the vertical direction;

k_(w) is the thermal conductivity of water;

k_(s) is the thermal conductivity of steel (or other material) formingthe tank;

ρ denotes density;

c denotes specific heat;

η denotes the efficiency or effectiveness of the heater, and ranges from0:1;

P is the heater power.

The unknown input estimators take on an arbitrary form of a second ordersystem for purpose of explanation as seen in Equations (17)-(20). Theassumed temperature measurements are T₁₀ and T₁₃ (so that measured v.estimated error quantities T ₁₀ and T ₁₃ are defined).

The skilled artisan can readily modify the illustrative water heater 20and/or model 33 of FIGS. 3-5 to model water heaters with differentnumbers of disks (thus providing different vertical resolution),different number or placement of temperature sensors and/or heaters, andso forth. Similarly, it is contemplated to omit or add additionalphysical characteristics to the model.

The illustrative model 33 can be used for various purposes. In theillustrative demand response example of FIG. 1, the model 33 can be usedto more accurately assess the energy stored in the tank at any giventime, so that the dispatch engine 26 can more effectively leverage theaggregate stored energy for demand response while ensuring thatindividual water heaters 20 can perform their intended function ofproviding hot water within certain constraints (temperature, capacity,et cetera). The energy stored in the tank 40 can be computed as the sumof the energy stored in each disk of the stacked disks representing thewater in the cylindrical tank 40. The energy stored in each disk can becomputed as Energy stored=C_(p)*dT*m where C_(p) is the specific heatcapacity of water, dT is the temperature difference between the water inthe disk and the surroundings, and m is mass of water.

In another illustrative application, the model 33 is used to providecondition based maintenance. The use of water heater sensors foridentification of maintenance issues affecting the water heateradvantageously benefits the end-user (who typically owns and utilizesthe water heater) by detecting potential problems early, while they canbe repaired in a cost-effective manner, or while they can be resolved byreplacing the water heater before a catastrophic failure such as alarge-scale water leak occurs. Condition based maintenance as disclosedherein also enables the water heater to be kept in efficient operatingcondition, thus lowering energy costs for the end user. These incentivesadvantageously encourage the end-user to participate in the aggregationmanaged by the dispatch engine 26. Optionally, the aggregation maycharge a fee for providing this maintenance monitoring, thus generatingan additional revenue stream for the demand response system operators.

By way of the illustrative example of FIGS. 2-5, the state of a waterheater 20 comprises temperature measurements by two (or more)temperature sensors measuring the water temperature, along with theelectrical power being consumed as suitably measured by a clamp-onammeter or the like. The state measurements are combined with the model33 of the water heater 20 to obtain estimates for parameters describingthe water heater 20, such as the effective insulation thermalresistance, first hour delivery, effective volume, heating element powertransfer rate, and so forth. These parameters can be used for providingdemand response as already described, and additionally can be used toinfer the (possibly incipient) presence of possible failure modes so asto provide condition based maintenance (CBM) capability. Condition basedmaintenance provides a mechanism for maintaining the water heater basedupon its actual condition prior to failure, rather than performingmaintenance at specified service intervals or simply waiting for thedevice to fail outright. Implementing CBM has several positive impactson the end user and the maintenance provider. For the end user, thecosts of maintaining a device can be reduced by servicing only at theonset of an issue rather than at regular intervals. Maintenanceschedules are generated for typically usage patterns, but if the usagepattern for a particular water heater differs from typical, maintenancecould be performed too frequently increasing costs, or too infrequentlyleading to device failure and replacement and repair costs. The end useralso benefits from a reduced number of service interruptions becauseoperational issues are detected prior to the failure of a device, andthe water heater can be serviced without the disruption of a failure. Anadditional financial benefit for the user is that the water heater iskept operating closer to its peak efficiency, reducing energy costs.Because water heaters operate transparently in the absence of a failure(that is, the user is typically satisfied as long as hot water of thedesired temperature is being delivered), the end-user may be unawarethat the water heater is operating under reduced performance andefficiency, because the user experience has not changed and any increasein energy cost is gradual and can go unnoticed. The CBM systemsdisclosed herein notify the user of the onset of a failure mode, such as(with brief reference back to FIG. 2) an insulation disturbance 60,fouling or build-up 62 on a heating element 44, excessive sedimentbuild-up 64, or a rupture 66 in the drip tube 48 generating a cold waterbypass flow path 68. By identifying the failure mode with someparticularity, the end user is alerted of the problem early, and cantake corrective action.

As a further benefit, maintenance providers (for example, in the contextof a water heater service contract) can increase the level of serviceprovided to their customers by using CBM while also reducing theircosts. The level of service to customers can be enhanced by reducing thenumber of unneeded, schedule based service calls, while also reducingthe likelihood of a device being unexpectedly removed from service. Themaintenance costs can be reduced by scheduling work to be performedduring normal business hours, rather than off-hours requiringdifferential or overtime pay. This can be done because failures can beanticipated based on the output of the CBM system, thus the providerscan operate proactively instead of reactively. When technicians aredispatched to a site, a better understanding the service needed isalready available prior to their departure, as the CBM system outputprovides the service technician with identification of the likelyfailure mechanism before the service technician examines the waterheater. This knowledge reduces the risk of a technician being dispatchedwithout the proper parts or tools.

With returning reference to FIG. 1, CBM system may be implemented assoftware or firmware executed in conjunction with the model 33 by themicroprocessor or microcontroller of the load controller 30, and/or maybe implemented as software or firmware executed at the dispatch engine26 which communicates with the load controller 30. Implementing the CBMat the load controller 30 reduces bandwidth costs and spreadscomputational complexity across the load controllers 30. The CBM for awater heater 20 leverages the temporal recording of at least twotemperature sensors and the power consumed by the water heater. The twotemperature sensors may, for example (see illustrative FIG. 2), belocated near the upper and lower thermostats 52, 54 typically found onthe water heater. From these measurements, the state of the water heateris estimated, for example using the model 33, where state is a measureof the energy stored within the device, and power being consumed. Thewater heater typically has a minimum and maximum usable energy state.The minimum energy is when the temperature(s) are at the deadbandminimum and the maximum energy is when the temperature(s) are at thedeadband maximum. The power consumption state may be discrete powerlevels, based upon the size and number of heaters operating. Using thesestate values, parameter identification can be performed, fittingmeasurements to the model 33 of the water heater 20. The parameters tobe estimated may, for example, include the effective capacity (thevolume of water the tank contains), the thermal conductivity from thetank to ambient, the effective mixing rate, first hour deliverycapacity, and the heating element size. By monitoring these parametersover time and using other error detection methodologies, changes insystem operation signifying the onset of a failure mechanism (broadlydefined herein as encompassing the spectrum from catastrophic failure,e.g. a major water leak or complete cessation of water heating, tofailures that produce less deleterious effects such as reducedoperational efficiency).

With reference to FIG. 4, an illustrative CBM system is described, whichmay be implemented at the load controller 30 or the dispatch engine 26.In an operation 70, the temperature readings and input power aremeasured (corresponding to T₁₀, T₁₃, and u₁ and u₂ in the model 33 ofEquations (1)-(20)), and these inputs are applied to the model 33 togenerate the estimated water temperatures {circumflex over (T)} andestimated water flows {dot over ({circumflex over (m)})}. Theseinformation then serve as the inputs to various illustrative waterheater failure mode detectors 72, 74, 76, 78 as described next. Itshould be noted that a detected failure mode does not necessarilyrequire that the failure has actually occurred—rather, a failure modedetector may “anticipate” a failure by detecting incipient degradationof a type having a significant potential to lead to a failure of thedetected failure mode. A failure mode may also be detected early, thatis, at a point where performance has been compromised by the failuremode to some degree but has not yet reached a point where remedialaction (e.g. repair or replacement) is appropriate.

An illustrative embodiment of an illustrative insulation disturbancedetector 72 operates to detect an insulation disturbance. The efficiencyof the water heater 20 is dependent upon its thermal conductivity fromthe tank to ambient temperature. This thermal conductivity is limited bythermal insulation of the storage tank 40. Insulation by itself is notlikely to fail on its own, but external factors could damage insulationsuch as the presence of water, exposure to airflow, or disturbance bypeople and animals. The occurrence of any of these factors warrantsservicing. A user would likely not notice the presence of an insulationfailure until the next utility billing cycle, and even then may fail tonotice the resulting loss of efficiency if its onset is gradual. Theinsulation disturbance detector 72 suitably estimates the R-value, whichis the inverse of the thermal conductivity (k) estimated by the 1D waterheater model 33. For a uniform insulator, the R-value is the ratio ofthe temperature difference across the insulator and the heat flux (whichis the heat transfer per unit area per unit time. This can be written asR=ΔT/{dot over (Q)} where Δt is the temperature difference across theinsulator, {dot over (Q)} is the heat flux, and R is the R-value. In the1D model of FIG. 3, the average R-value can be computed for each annularsegment 10, 11, 12, 13, 14, 15, 16 of the cylindrical wall of thevertically oriented cylindrical water storage tank 40. Since the R-valueis the inverse of thermal conductivity (k), the computation canalternatively be done in terms of k and lumped into the heat transfercoefficient with the environment, U. In one approach, the thermalconductivity is estimated and stored daily (or on some other interval,e.g. weekly, monthly, et cetera). An insulation disturbance is suitablydetected as a large change in the day to day (or interval-to-interval)R-value, or as a predefined percent change, from the initial valuemeasured at installation of the water heater 20. The first case (asudden change in R-factor) is typically due to disruptions in theinsulation from external factors, such as a pipe leak above the waterheater which moistens the insulation, causing its failure. The secondcase (more gradual change I R-factor) is indicative of insulationdegradation over time, such as due to environmental conditions ormanufacturing errors that are causing the insulation to slowly degrade.Depending upon the chosen predefined percent change (or other chosenthreshold), the insulation disturbance detector 72 may operate to detectinsulation failure in an incipient stage, i.e. before it is severeenough to call for maintenance or replacement.

An illustrative embodiment of an illustrative heating element failuredetector 74 operates to detect fouling or build-up on a heating elementthat can reduce its heat transfer efficiency. The heating element 42, 44is an electrically resistive device that dissipates power into the waterin the tank 40. An ideal heating element would dissipate all of itssupplied power into the water instantaneously. However, existing heatingelements are not ideal, and there will be some resistance to heattransfer from the heating element to the water. The resistance of theheating element, and therefore its power consumption, depend upon itsoperating temperature. Using these properties, the effectiveness of theheater can be identified, and changes in its effectiveness noted. Aheating element that is becoming fouled is expected to have reducedability to transfer heat to the water. This is because the media foulingthe heating element will likely have lower thermal conductivity comparedto the water itself. With reduced thermal conductivity, the rate oftemperature rise of the water due to action of the heating element willbe diminished, while the temperature of the heating element itself forthe same conditions will increase because heat is not being transferredto the water as effectively. Two parameter estimates can be used todetect this failure mode: the effective capacity of the tank; and theeffective electrical resistance of the heating element. Note that theeffective capacity can be considered as either a purely thermal quantity(e.g. number of Joules that can be stored) or as a volumetric quantitysince each unit of water has a certain thermal capacity. The effectivecapacity of the tank will appear to increase if the heating element isbecoming fouled because a slower temperature rise for the same powerinput will be observed. The rate of temperature rise is dependent uponthe power input and the volume of water to be heated, which can bewritten as Q=MCΔt where Q is the energy imparted into the system, M isthe volume of water, C is the heat capacity of water, and Δt is thetemperature rise. The value of M should be constant over all time for awater heater 20, or nearly so. If the heating element is failing due tofouling or buildup, then the same heat Q, which is the electrical powerintegrated over time, will result in a different Δt, signifying a changein the other independent variable, M. The temperature rise Δt willdecrease, implying an increase in M. Analogously to the situation forinsulation failure, both short term differences in M and long termdifferences in M may be monitored to determine if either a dramatic orgradual failure is occurring.

In an alternative embodiment, heating element failure due to fouling orbuildup is detected by estimating the electrical resistance of theheating element. In a suitable approach, the resistance of the heatingelement may be measured using V=I·R where V is the voltage, I is theelectrical current (both V and I suitably being represented as root meansquared, or RMS, values), and R is the resistance. The resistance willshow a temperature dependency, thus it is possible to detect if theheating element is operating at a much higher or lower temperature thenprescribed. Again, the effective resistance would be compared to bothshort term and long term data to identify if a rapid or gradual failureis occurring.

A second heating element failure mode optionally detected by the heatingelement failure detector 74 is the development of areas with lowelectrical conductivity, or cold spots, on the heating element. When acold spot develops, the effective electrical resistance will increase.The previous methods can be employed to detect this failure type bydetermining if there was an increase in the effective resistance.

An illustrative embodiment of an illustrative excessive sediment buildupdetector 76 operates to detect excessive sediment build-up 64 thatreduces the actual water capacity of the tank 40. Sediment from thewater supply may become entrapped within the storage tank 40. Somesediment build up is to be expected, and an anode rod (not shown in FIG.2) is included in some water heater designs to suppress sedimentbuildup. However excessive sediment buildup remains a possibility evenwith anode rod protection, and excessive sediment buildup will result ina decrease in performance. In extreme cases, sediment 64 can occupy avolume inside the tank 40 which is large enough to cause a decrease inperformance as experienced by the end user. With excessive sedimentbuildup, the storage tank 40 does not contain its rated capacity,limiting the volume of heated water available. The presence of thismagnitude of sediment typically justifies servicing the water heater 20,as well as examining the water supply for more serious issues which maybe the cause of excess sediment buildup. In one approach, the detector76 suitably detects sediment buildup by calculating the effectivecapacity of the water tank 40, and comparing it to the capacitycalculated at installation. Sediment buildup occurs slowly over time,and a maintenance issue is suitably reported if the measured tankcapacity falls below a specified percentage of the starting capacity,which may be variously chosen to choose the level of sediment buildup tobe detected (ranging from reporting incipient buildup to major buildup).Multiple thresholds are optionally used, e.g. an “incipient problem”threshold that is triggered for a small reduction in measured tankcapacity, and a “serious problem” threshold that is triggered only ifthe measured tank capacity is more significantly reduced.

It will be observed that there are some similarities between thesymptoms of excessive sediment buildup and the symptoms of a failedheating element, and the detectors 74, 76 are preferably configured todistinguish between these two failure modes. In general, both heatingelement failure and excessive sediment buildup manifest as an apparentchange in capacity of the tank 40. However, sediment buildup cause aperceived decrease in tank capacity; whereas, heating element failurecauses a perceived increase in tank capacity. Thus, the direction ofchange in apparent tank capacity output by the model 33 is suitably usedby the detectors 74, 76 to distinguish between the heating elementfailure and sediment buildup failure modes.

An illustrative embodiment of an illustrative drip tube rupture detector78 operates to detect a rupture 66 in the drip tube 48. As seen in FIG.2, the drip tube 48 supplies unheated (i.e. cold) water to the bottom ofthe water storage tank 40, and the heated water rises to the outlet 50at the top of the tank 40. This flow path design ensures that the drawnwater is hot water rather than newly injected cold water. However, if arupture 66 forms in the drip tube 48, this can produce a cold waterbypass flow 68 via which unheated (i.e. cold) supply water can passdirectly to the outlet 50 resulting in lower effectiveness andefficiency of the water heater. A drip tube failure may requirereplacement of the entire tank or water heater, so rapid diagnosis ofthis problem can avoid unnecessary attempts at remedial maintenance ofthe water heater. The detector 78 suitably detects the drip tube rupture66 by monitoring the difference in temperature readings of thetemperature sensors located at upper and lower positions on the watertank 40, e.g. the difference in temperature readings for the upper andlower thermostats 52, 54 in the illustrative example of FIG. 2. Undernormal operation, cool water enters at the bottom of the tank via thedrip tube 48 in response to the drawing of hot water via the outlet 50,and heated water at lower elevation in the tank rises toward the top toreplace the drawn hot water. During hot water draw from the tank 40, thetemperature readings at the thermostats 52, 54 during normal operationwill reflect this, with the lower temperature reading starting todecrease before the upper temperature reading starts to decrease. Bycontrast, if the drip tube 48 includes a rupture 66 generating a bypasspath 68, then the upper temperature reading will begin to fall beforethe lower temperature reading, and may even fall below the lowertemperature reading, because the cool water is being fed in close to thetop of the tank. Such measurements are suitably performed on every usagecycle (triggered by detection of a rapid temperature decrease due to thecold water injection). While this temperature effect can be observedwith limited resolution directly from the temperature readings of thetwo temperature sensors, it is more accurately observed using the waterheater model, i.e. in the absence of a drip tube rupture the temperatureof the lower-elevation disks should drop faster than the temperature ofthe higher-elevation disks; whereas, if a cold water bypass exists dueto a drip tube rupture then to the contrary the temperature of thehigher-elevation disks should drop faster than the temperature of thelower-elevation disks.

With continuing reference to FIG. 4, the diagnostic outputs of thefailure mode detectors 72, 74, 76, 78 are suitably processed by a CBMreporting module 80 that generates a human-perceptible report of anydetected failure, or if no failure is detected then a human-perceptiblereport that no failure has been detected. Because there is a relativelysmall finite set of failure modes being monitored, in some embodimentsthe CBM reporting module 80 includes a memory or database storingnatural language (e.g. English) text describing each potential failure(or lack thereof) along with the detected symptoms as described above.Such a report may be transmitted to the facility maintenance entity 34,for example as a notification push and/or as a transmitted electronicmail (email) message or so forth. Additionally or alternatively, the CBMreporting module may report any detected failure mode (or lack thereof)at a lower-level format, for example transmitting a failure diagnosisbinary string of N bits to the dispatch engine 26 where (by way ofillustration) bit zero is set to “0” if there is no insulationdisturbance and is set to “1” if there is a detected insulationdisturbance, and similarly (using bits one, two, . . . of the binarystring) for the remaining failure modes that are monitored by the CBMsystem. In general, the processing components 33, 72, 74, 76, 78, 80 aresuitably implemented as software or firmware executed by themicrocontroller or microprocessor of the load controller 30 and/or bythe dispatch engine 26.

The CBM system has been described as operating in conjunction with inconjunction with the illustrative demand response system described withreference to FIG. 1, and in conjunction with the illustrative electricwater heater(s) 20. However, it will be appreciated that the CBM systemcan be a standalone system not operating in conjunction with any demandresponse system, or can be an ancillary system operating in conjunctionwith some other type of automated water heater control system (e.g., asystem coordinating operation of a bank of water heaters supplying hotwater to a common outlet). It will be further appreciated that thedisclosed CBM system can operate to monitor failure modes of other typesof loads. For example, adaptation to CBM monitoring of a natural gas (orother gas-fired) water heater is straightforward, as the resistiveheating elements 42, 44 are replaced by a gas line and the power inputmonitoring (e.g. by a clamp-on ammeter in the case of an electric waterheater) is suitably replaced by gas flow monitoring in combination witha suitable conversion factor converting gas flow to power input. Theinsulation disturbance, excessive sediment, and drip tube rupturefailure modes can also occur in a gas-fired water heater, while thefailure mode of resistive heating element fouling or cold spot evolutiontranslates to failure modes that compromise operation of the gas burner.

Still further, it is contemplated to employ the disclosed CBM systemsand methods in conjunction with loads other than water heaters. By wayof illustrative example, CBM of heating, ventilation, and airconditioning (HVAC) systems entails modeling HVAC operation based oninputs including the electrical power input to the HVAC system, roomtemperature, outside temperature, and air conditioner duty cycle todetect changes in HVAC performance, capacity, and efficiency. A failuremode such as blower resistance due to filter blockage can be detectedbased on increased blower electrical current draw to force air throughthe partially blocked filter, thus enabling a condition-basednotification or email that the filter should be replaced. A refrigerantleak can be detected by observing normal blower operation and normalHVAC duty cycling in conjunction with less efficient cooling as observedby a less rapid temperature drop when the HVAC is operating, or a higherduty cycle overall to maintain the temperature set point. In amulti-room HVAC system, duct occlusion can be detected as reducedcooling efficiency for one room as compared with other rooms cooled bythe HVAC system, from which an occlusion of the duct feeding that oneroom can be inferred.

It may be noted that the diagnostic accuracy of the CBM system may beless than perfect. This is because the CBM system is providing adiagnostic aid, but typically does not perform the diagnosis upon whichmaintenance decisions are made (much less the physical maintenance),both of which remain in the domain of human maintenance personnel.Rather, the CBM system provides an indication that a certain failuremode may be present, calling for investigation by human maintenancepersonnel. Thus, so long as the CBM system provides sufficientdiagnostic accuracy (for example, as measured by a false positives ratein which a failure is detected that is ultimately determined to not bepresent, and/or by a false negatives rate in which a failure that ispresent is not detected by the CBM system) so that maintenance costsoverall are reduced, or overall operational efficiency is increased,then the CBM system provides a useful benefit, such as providing anancillary benefit to end users incentivizing (at least in part)participation in the demand response aggregation.

The preferred embodiments have been illustrated and described.Obviously, modifications and alterations will occur to others uponreading and understanding the preceding detailed description. It isintended that the invention be construed as including all suchmodifications and alterations insofar as they come within the scope ofthe appended claims or the equivalents thereof.

The invention claimed is:
 1. A water heater control system forcontrolling a water heater including a vertically oriented cylindricalwater storage tank having a cylindrical wall, at least one heatingelement arranged to heat water in the water storage tank, and at leastone temperature sensor arranged to measure water temperature in thewater storage tank, the electric water heater control system comprising:a load controller comprising an electronic data processing deviceconfigured to operate the water heater including operating the at leastone heating element based on temperature readings provided by the atleast one temperature sensor to control water temperature of water inthe water storage tank, wherein the load controller is furtherconfigured to (i) communicate with an aggregation demand responsedispatch engine comprising an electronic data processing device that isconfigured to send demand response commands to an aggregation of loadsincluding the water heater and (ii) operate the water heater inaccordance with demand response commands received from the aggregationdemand response dispatch engine, to perform a demand response operation;and a water heater modeling component comprising an electronic dataprocessing device configured to model the water heater using aone-dimensional model that includes: a vertical stack of disksrepresenting the water volume in the cylindrical water storage tank, anda stack of annular segments surrounding the vertical stack of diskswherein the stack of annular segments represents the cylindrical wall ofthe cylindrical water storage tank, wherein the demand responseoperation, performed by the load controller and aggregation demandresponse dispatch engine, is based at least in part on energy stored inthe vertically oriented cylindrical water storage tank as determinedusing the water heater modeling component.
 2. The water heater controlsystem of claim 1 further comprising: a condition-based maintenancesystem comprising an electronic data processing device configured todetect a failure mode present in the water heater based on an output ofthe water heating model component.
 3. The water heater control system ofclaim 2 wherein the condition-based maintenance system is configured todetect a failure mode comprising insulation disturbance present in thewater heater based on R-values or thermal conductivity values computedfor the annular segments representing the cylindrical wall of thecylindrical water storage tank.
 4. The water heater control system ofclaim 2 wherein the condition-based maintenance system is configured todetect a failure mode comprising a heating element failure present inthe water heater based on an increase over time of the thermal orvolumetric capacity of the water in the water storage tank determinedusing the water heater modeling component.
 5. The water heater controlsystem of claim 2 wherein the water heater is an electric water heater,the at least one heating element is a resistive heating element, and thecondition-based maintenance system is configured to detect a failuremode comprising a heating element failure present in the water heaterbased on an increase in electrical resistance measured for the heatingelement over time.
 6. The water heater control system of claim 2 whereinthe condition-based maintenance system is configured to detect a failuremode comprising sediment buildup present in the water storage tank ofthe water heater based on a decrease over time of the thermal orvolumetric capacity of the water in the water storage tank determinedusing the water heater modeling component.
 7. The water heater controlsystem of claim 2 wherein the condition-based maintenance system isconfigured to detect a failure mode comprising a drip tube rupturepresent in the water heater based on more rapid cooling of the upperelevation disks of the vertical stacked disks as compared with the lowerelevation disks during a hot water draw event.
 8. The water heatercontrol system of claim 1 wherein the one-dimensional model comprisescoupled differential equations including: for each disk of the verticalstack of disks representing the water volume in the cylindrical waterstorage tank, a differential equation expressing the time derivative ofthe temperature of the disk estimated by the one-dimensional model; andfor each annular segment of the stack of annular segments representingthe cylindrical wall of the cylindrical water storage tank, adifferential equation expressing the time derivative of the temperatureof the annular segment estimated by the one-dimensional model.
 9. Thewater heater control system of claim 1 wherein the load controller andthe water heater modeling component are constructed as a singleelectronic data processing device configured to both operate the waterheater and model the water heater using the one-dimensional model.
 10. Asystem comprising a water heater including a water storage tank, atleast one heating element arranged to heat water in the water storagetank, and at least one temperature sensor arranged to measure watertemperature in the water storage tank; a load controller comprising anelectronic data processing device configured to operate the water heaterincluding operating the at least one heating element based ontemperature readings provided by the at least one temperature sensor tocontrol water temperature of water in the water storage tank, whereinthe load controller is further configured to (i) communicate with anaggregation demand response dispatch engine comprising an electronicdata processing device that is configured to send demand responsecommands to an aggregation of loads including the water heater and (ii)operate the water heater in accordance with demand response commandsreceived from the aggregation demand response dispatch engine, toperform a demand response operation; a water heater modeling componentcomprising an electronic data processing device configured to model thewater heater using a one-dimensional model that includes: a verticalstack of disks representing the water volume in the cylindrical waterstorage tank, and a stack of annular segments surrounding the verticalstack of disks wherein the stack of annular segments represents thecylindrical wall of the cylindrical water storage tank; and acondition-based maintenance system comprising an electronic dataprocessing device configured to detect a failure mode present in thewater heater based on information including the temperature readingsprovided by the at least one temperature sensor and power input to thewater heater, wherein the demand response operation, performed by theload controller and aggregation demand response dispatch engine, isbased at least in part on energy stored in the vertically orientedcylindrical water storage tank as determined using the water heatermodeling component.
 11. The system of claim 10, wherein the water heatermodeling component comprising the electronic data processing device isfurther configured to model the water heater using the one-dimensionalmodel of the water heater that receives as inputs the temperaturereadings provided by the at least one temperature sensor and power inputto the water heater, wherein the condition-based maintenance system isconfigured to detect a failure mode present in the water heater based onparameter values estimated by the water heater modeling component. 12.The system of claim 10 wherein the condition-based maintenance system isconfigured to detect a failure mode comprising insulation disturbancepresent in the water heater based on R-values or thermal conductivityvalues computed for the annular segments representing the wall of thewater storage tank.
 13. The system of claim 11 wherein thecondition-based maintenance system is configured to detect a failuremode comprising a heating element failure present in the water heaterbased on an increase over time of the thermal or volumetric capacity ofthe water in the water storage tank determined using the water heatermodeling component.
 14. The system of claim 11 wherein thecondition-based maintenance system is configured to detect a failuremode comprising sediment buildup present in the water storage tank ofthe water heater based on a decrease over time of the thermal orvolumetric capacity of the water in the water storage tank determinedusing the water heater modeling component.
 15. The system of claim 10wherein the water heater is an electric water heater, the at least oneheating element is a resistive heating element, and the condition-basedmaintenance system is configured to detect a failure mode comprising aheating element failure present in the water heater based on an increasein electrical resistance measured for the heating element over time. 16.The system of claim 10 wherein the water heater included upper and lowertemperature sensors measuring temperature in an upper portion of thewater storage tank and in a lower portion of the water storage tank,respectively, and the condition-based maintenance system is configuredto detect a failure mode comprising a drip tube rupture present in thewater heater based on more rapid decrease in temperature read by theupper temperature sensor as compared with temperature read by the lowertemperature sensor during a hot water draw event.